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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 3249.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3249.g1 | 3249e3 | \([1, -1, 0, -329841, 72994702]\) | \(115714886617/1539\) | \(52782232316211\) | \([2]\) | \(17280\) | \(1.7765\) | |
3249.g2 | 3249e2 | \([1, -1, 0, -21186, 1078087]\) | \(30664297/3249\) | \(111429157112001\) | \([2, 2]\) | \(8640\) | \(1.4299\) | |
3249.g3 | 3249e1 | \([1, -1, 0, -4941, -114296]\) | \(389017/57\) | \(1954897493193\) | \([2]\) | \(4320\) | \(1.0834\) | \(\Gamma_0(N)\)-optimal |
3249.g4 | 3249e4 | \([1, -1, 0, 27549, 5279044]\) | \(67419143/390963\) | \(-13408641905810787\) | \([2]\) | \(17280\) | \(1.7765\) |
Rank
sage: E.rank()
The elliptic curves in class 3249.g have rank \(1\).
Complex multiplication
The elliptic curves in class 3249.g do not have complex multiplication.Modular form 3249.2.a.g
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.